Abstract
Orthogonal wavelet bases are not optimal because they cannot capture all forms of regularity in a signal. In images, large wavelet coefficients appear along contours, but these coefficients do not capture the geometric regularity of the contour. A basis adapted to the geometry must be used to improve the nonlinear approximation and thus the estimation of the image in the presence of noise. This has been achieved using curvelet and bandlet bases. However, these are still not as effective as noise-removal estimators based on neural networks.
We demonstrate that the score estimator using an unbiased neural network is equivalent to thresholding in an orthonormal basis whose choice depends on the noisy signal. The vectors of this basis are the eigenvectors of the Hessian of the log-likelihood of the noisy signal. Numerical experiments show that these bases generalize the bases studied in harmonic analysis and adapt to the image’s geometry. For classes of geometrically regular images, we observe that the estimators computed by neural networks produce errors whose asymptotic decay is optimal. However, the mathematical properties of these orthonormal bases remain poorly understood.
